EXTREME POINTS OF THE UNIT BALL OF AN OPERATOR SPACE MASAYOSHI KANEDA* Dedicated to the memory of Gert K. Pedersen Abstract. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) “quasi-identities”. Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a C∗-algebra or a one-sided ideal in a C∗-algebra in terms of quasi-multipliers. An extreme point is also used to show that any TRO with predual can be decomposed to the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra. 1. Introduction. Quasi-multipliers of operator spaces were introduced by V. I. Paulsen ([19] Definition 2.2) in late 2002 as a natural variant of one-sided multipliers of operator spaces which had been introduced and studied by D. P. Blecher around 1999 ([1]). However, the significant corre- spondence of quasi-multipliers and operator algebra products was discovered by the author ([19] Theorem 2.6) in early 2003. That is, for a given operator space X, the possible opera- tor algebra products that X can be equipped with are precisely the bilinear mappings on X that are implemented by contractive quasi-multipliers. Moreover, in [17] Theorem 3.3.1 and [18] Theorem 4.1, the author gave a geometric characterization of operator algebra products in terms of only matrix norms using the Haagerup tensor product. These results were pre- sented in the Great Plains Operator Theory Symposium (GPOTS) held at the University of Illinois at Urbana-Champaign in 2003. After the author’s talk, G. K. Pedersen asked the author the question “How can the extreme points of the unit ball of a quasi-multiplier space be characterized?” This question gave the author a further direction to study about arXiv:math/0408235v2 [math.OA] 18 May 2009 quasi-multipliers. Through investigation, it turned out that what should be characterized is the “interlocking” between quasi-multipliers and extreme points of the unit ball of the Date: August 31, 2018. Mathematics subject classification 2000. Primary 47L07; Secondary 47L30, 47L25, 46L07, 47L50, 46M10, 46L05. Key words and phrases. quasi-multipliers, extreme points, abstract operator algebras, approximate iden- tities, injective operator spaces, dual operator spaces, C∗-algebras, ideals, ternary rings of operators. THIS PAPER IS A REVISION AND AN ENLARGEMENT OF THE AUTHOR’S MANUSCRIPT TI- TLED “EXTREME POINTS OF THE UNIT BALL OF A QUASI-MULTIPLIER SPACE” WHICH HAD BEEN CIRCULATED SINCE 2004. * The author was supported by a research fund from Department of Mathematics, University of California, Irvine. 1 2 MASAYOSHI KANEDA* operator space, and not extreme points of the unit ball of quasi-multiplier space alone. For this sake, the author introduces the new notion (approximate) “quasi-identities”. In Section 2 we briefly review a construction of injective envelopes and triple envelopes of an operator space, and recall the definition of quasi-multipliers and their correspondence to operator algebra products. Furthermore, we define important classes of extreme points: local isometries, local co-isometries, and local unitaries, which actually becomes isometries, co-isometries, and unitaries, respectively, in certain cases with certain embeddings. In Section 3 we give alternative definitions of one-sided and quasi-multipliers which are used to characterize operator algebras with approximate identities in Section 4. In Section 4 we introduce the new notion: (approximate) “quasi-identities” for normed algebras. We see that at least in the operator algebra case, contractive (approximate) quasi- identities are natural generalization of contractive (approximate) one-sided identities. Then we characterize an operator algebra with a contractive (approximate) quasi- (respectively, left, right, two-sided) identity in terms of its associated quasi-multiplier and extreme points of the unit ball of (the weak∗-closure of) the underlying operator space. In Section 5 we give an operator space characterization of C∗-algebras and their one-sided ideals in a very clear manner in terms of quasi-multipliers. Section 6 is devoted to showing that if an operator space has an operator space predual, then so is its quasi-multiplier space. In Section 7 it is shown that any TRO with predual can be decomposed to the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra using an extreme point of the TRO. This paper is a revision and an enlargement of the author’s manuscript titled “Extreme points of the unit ball of an quasi-multiplier space” which had been circulated since 2004. The author thanks David P. Blecher for pointing out a gap in the initial manuscript. Portion of the work was carried out while the author was a post-doctoral researcher at the University of California, Irvine. The author is grateful to Bernard Russo for his invitation, financial support, and warm hospitality. As stated before, this work was motivated by Pedersen’s question posed to the author. However, he passed away within a year of his asking the question. We regret that our answer to his question did not make it while he was alive. Henceforth the author would like to dedicate this paper as a requiem to the memory of Gert K. Pedersen. 2. Preliminaries. We begin by recalling the construction of an injective envelope of an operator space due to Z.-J. Ruan ([23], [24]) and M. Hamana ([13], [14], also see [15]), independently. The reader unfamiliar with this subject is referred to [21] Chapter 15, [11] Chapter 6, [7], or [1], for example. Let X ⊂ B(H) be an operator space, and consider Paulsen’s operator system C1 X S := H ⊂ M (B(H)). X X∗ C1 2 H EXTREME POINTS OF THE UNIT BALL OF AN OPERATOR SPACE 3 One then takes a minimal (with respect to a certain ordering) completely positive SX - 1 projection Φ on M2(B(H)), whose image ImΦ turns out to be an injective envelope I(SX ) ∗ of SX . By a well-known result of M.-D. Choi and E. G. Effros ([8]), ImΦ is a unital C -algebra with the product ⊙ (which is called the Choi-Effros product) defined by ξ ⊙ η := Φ(ξη) for ξ, η ∈ImΦ and with other algebraic operations and norm taken to be the original ones in ∗ M2(B(H)). The C -algebra structure of I(SX ) does not depend on the particular embedding X ⊂ B(H). By a well-known trick one may decompose to ψ φ Φ= 1 . φ∗ ψ 2 Accordingly, one may write I (X) I(X) (1) ImΦ= I(S )= 11 ⊂ M (B(H)), X I(X)∗ I (X) 2 22 ∗ where I(X) is an injective envelope of X, and I11(X) and I22(X) are injective C -algebras (hence unital (See [7] Proposition 2.8.)). We denote the identities of I11(X) and I22(X) by 111 and 122, respectively. Note that the last inclusion in Expression (1) is not as a subalgebra since the multiplication in I(SX ) and the multiplication in M2(B(H)) are not same in general. The new product ⊙ induces a new product • between elements of I11(X), I22(X), I(X), and ∗ ∗ ∗ I(X) . For instance, x • y = ψ1(xy ) for x, y ∈ I(X). Note that the associativity of • is guaranteed by that of ⊙. The following property is often useful. Lemma 2.1. (Blecher-Paulsen [7] Corollary 1.3) (1) If a ∈ I11(X), and if a • x =0, ∀x ∈ X, then a =0. (2) If b ∈ I22(X), and if x • b =0, ∀x ∈ X, then b =0. One may write the C∗-subalgebra C∗(∂X) of ImΦ (with the new product) generated by O X O O as E(X) T (X) I (X) I(X) C∗(∂X)= ⊂ 11 , T (X)∗ F(X) I(X)∗ I (X) 22 where T (X) is a triple envelope of X, i.e., a “minimum” TRO that contains X completely isometrically. Here an operator space X being a ternary ring of operators (TRO for short) or a triple system means that there is a complete isometry ι from X into a C∗- algebra such that ι(x)ι(y)∗ι(z) ∈ ι(X), ∀x, y, z ∈ X. O X We call the embedding i : X → ⊂ C∗(∂X) ⊂ I(S ) the Silovˇ embedding of O O X 0 x O X X, and often denote (x ∈ X) simply by x. Similarly, we often write X for , 0 0 O O 1 An SX -projection is an idempotent that fixes each element in SX . 4 MASAYOSHI KANEDA* 1 0 and 1 for 11 , etc. The involution in B(H) induces an involution in M (B(H)) in an 11 0 0 2 ∗ 0 x 0 0 obvious way, and we still denote by ∗. For example, for x ∈ X, = . 0 0 x∗ 0 In this paper, all operator spaces are assumed to be norm-closed . Whenever an infinite- dimensional vector space is involved in a product, we take the norm closure of the linear span. For instance, X • z • X := span{x • z • y; x, y ∈ X}, where z ∈ I(X)∗. Now we are ready to recall the definition of one-sided and quasi-multipliers. We remark that the one-sided multipliers were first introduced by D. P. Blecher in [1]. The following definition (Items (1) and (2)) is an equivalent but more manageable version in [7]. Definition 2.2. Let X be an operator space.